\(\int \frac {(A+B x) (d+e x)^{7/2}}{(b x+c x^2)^{5/2}} \, dx\) [1278]

Optimal result

Integrand size = 28, antiderivative size = 524 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {2 (d+e x)^{5/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b c d^2 \left (8 A c^2 d+b^2 B e-b c (4 B d+9 A e)\right )+\left (16 A c^4 d^3-4 b^4 B e^3+b^3 c e^2 (4 B d+A e)-8 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (5 B d+6 A e)\right ) x\right )}{3 b^4 c^2 \sqrt {b x+c x^2}}-\frac {2 \left (16 A c^4 d^3-8 b^4 B e^3+b^3 c e^2 (5 B d+2 A e)-8 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (5 B d+4 A e)\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} c^{5/2} \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 d (c d-b e) \left (16 A c^3 d^2+4 b^3 B e^2+b^2 c e (B d-A e)-8 b c^2 d (B d+2 A e)\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{3 (-b)^{7/2} c^{5/2} \sqrt {d+e x} \sqrt {b x+c x^2}} \]

[Out]

-2/3*(e*x+d)^(5/2)*(A*b*c*d+(2*A*c^2*d+b^2*B*e-b*c*(A*e+B*d))*x)/b^2/c/(c*x^2+b*x)^(3/2)+2/3*(b*c*d^2*(8*A*c^2
*d+b^2*B*e-b*c*(9*A*e+4*B*d))+(16*A*c^4*d^3-4*b^4*B*e^3+b^3*c*e^2*(A*e+4*B*d)-8*b*c^3*d^2*(3*A*e+B*d)+b^2*c^2*
d*e*(6*A*e+5*B*d))*x)*(e*x+d)^(1/2)/b^4/c^2/(c*x^2+b*x)^(1/2)-2/3*(16*A*c^4*d^3-8*b^4*B*e^3+b^3*c*e^2*(2*A*e+5
*B*d)-8*b*c^3*d^2*(3*A*e+B*d)+b^2*c^2*d*e*(4*A*e+5*B*d))*EllipticE(c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e/c/d)^(1/2))
*x^(1/2)*(1+c*x/b)^(1/2)*(e*x+d)^(1/2)/(-b)^(7/2)/c^(5/2)/(1+e*x/d)^(1/2)/(c*x^2+b*x)^(1/2)+2/3*d*(-b*e+c*d)*(
16*A*c^3*d^2+4*b^3*B*e^2+b^2*c*e*(-A*e+B*d)-8*b*c^2*d*(2*A*e+B*d))*EllipticF(c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e/c
/d)^(1/2))*x^(1/2)*(1+c*x/b)^(1/2)*(1+e*x/d)^(1/2)/(-b)^(7/2)/c^(5/2)/(e*x+d)^(1/2)/(c*x^2+b*x)^(1/2)

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 524, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {832, 857, 729, 113, 111, 118, 117} \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\frac {2 d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \left (b^2 c e (B d-A e)-8 b c^2 d (2 A e+B d)+16 A c^3 d^2+4 b^3 B e^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{3 (-b)^{7/2} c^{5/2} \sqrt {b x+c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (b^3 c e^2 (2 A e+5 B d)+b^2 c^2 d e (4 A e+5 B d)-8 b c^3 d^2 (3 A e+B d)+16 A c^4 d^3-8 b^4 B e^3\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} c^{5/2} \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 (d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b c d^2 \left (-b c (9 A e+4 B d)+8 A c^2 d+b^2 B e\right )+x \left (b^3 c e^2 (A e+4 B d)+b^2 c^2 d e (6 A e+5 B d)-8 b c^3 d^2 (3 A e+B d)+16 A c^4 d^3-4 b^4 B e^3\right )\right )}{3 b^4 c^2 \sqrt {b x+c x^2}} \]

[In]

Int[((A + B*x)*(d + e*x)^(7/2))/(b*x + c*x^2)^(5/2),x]

[Out]

(-2*(d + e*x)^(5/2)*(A*b*c*d + (2*A*c^2*d + b^2*B*e - b*c*(B*d + A*e))*x))/(3*b^2*c*(b*x + c*x^2)^(3/2)) + (2*
Sqrt[d + e*x]*(b*c*d^2*(8*A*c^2*d + b^2*B*e - b*c*(4*B*d + 9*A*e)) + (16*A*c^4*d^3 - 4*b^4*B*e^3 + b^3*c*e^2*(
4*B*d + A*e) - 8*b*c^3*d^2*(B*d + 3*A*e) + b^2*c^2*d*e*(5*B*d + 6*A*e))*x))/(3*b^4*c^2*Sqrt[b*x + c*x^2]) - (2
*(16*A*c^4*d^3 - 8*b^4*B*e^3 + b^3*c*e^2*(5*B*d + 2*A*e) - 8*b*c^3*d^2*(B*d + 3*A*e) + b^2*c^2*d*e*(5*B*d + 4*
A*e))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*(
-b)^(7/2)*c^(5/2)*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (2*d*(c*d - b*e)*(16*A*c^3*d^2 + 4*b^3*B*e^2 + b^2*c*
e*(B*d - A*e) - 8*b*c^2*d*(B*d + 2*A*e))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]
*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*(-b)^(7/2)*c^(5/2)*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

Rule 111

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2*(Sqrt[e]/b)*Rt[-b/
d, 2]*EllipticE[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[
d*e - c*f, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !LtQ[-b/d, 0]

Rule 113

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Dist[Sqrt[e + f*x]*(Sqrt[
1 + d*(x/c)]/(Sqrt[c + d*x]*Sqrt[1 + f*(x/e)])), Int[Sqrt[1 + f*(x/e)]/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]), x], x] /
; FreeQ[{b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] &&  !(GtQ[c, 0] && GtQ[e, 0])

Rule 117

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2/(b*Sqrt[e]))*Rt
[-b/d, 2]*EllipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] &&
GtQ[c, 0] && GtQ[e, 0] && (PosQ[-b/d] || NegQ[-b/f])

Rule 118

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Dist[Sqrt[1 + d*(x/c)]*
(Sqrt[1 + f*(x/e)]/(Sqrt[c + d*x]*Sqrt[e + f*x])), Int[1/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]*Sqrt[1 + f*(x/e)]), x],
x] /; FreeQ[{b, c, d, e, f}, x] &&  !(GtQ[c, 0] && GtQ[e, 0])

Rule 729

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[Sqrt[x]*(Sqrt[b + c*x]/Sqrt[b
*x + c*x^2]), Int[(d + e*x)^m/(Sqrt[x]*Sqrt[b + c*x]), x], x] /; FreeQ[{b, c, d, e}, x] && NeQ[c*d - b*e, 0] &
& NeQ[2*c*d - b*e, 0] && EqQ[m^2, 1/4]

Rule 832

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(d + e*x)^(m - 1))*(a + b*x + c*x^2)^(p + 1)*((2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*
g - c*(b*e*f + b*d*g + 2*a*e*g))*x)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Dist[1/(c*(p + 1)*(b^2 - 4*a*c)), Int[(d
+ e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Simp[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2
*a*e*(e*f*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*(m + p + 1) + 2*c^2*d*f*(m
+ 2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2*p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] &
& RationalQ[a, b, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

Rule 857

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 (d+e x)^{5/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac {2 \int \frac {(d+e x)^{3/2} \left (\frac {1}{2} d \left (4 b B c d-8 A c^2 d-b^2 B e+9 A b c e\right )+\frac {1}{2} e \left (2 A c^2 d+4 b^2 B e-b c (B d+A e)\right ) x\right )}{\left (b x+c x^2\right )^{3/2}} \, dx}{3 b^2 c} \\ & = -\frac {2 (d+e x)^{5/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b c d^2 \left (8 A c^2 d+b^2 B e-b c (4 B d+9 A e)\right )+\left (16 A c^4 d^3-4 b^4 B e^3+b^3 c e^2 (4 B d+A e)-8 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (5 B d+6 A e)\right ) x\right )}{3 b^4 c^2 \sqrt {b x+c x^2}}+\frac {4 \int \frac {-\frac {1}{4} b d e \left (8 A c^3 d^2-4 b^3 B e^2+b^2 c e (2 B d+A e)-b c^2 d (4 B d+11 A e)\right )-\frac {1}{4} e \left (16 A c^4 d^3-8 b^4 B e^3+b^3 c e^2 (5 B d+2 A e)-8 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (5 B d+4 A e)\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3 b^4 c^2} \\ & = -\frac {2 (d+e x)^{5/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b c d^2 \left (8 A c^2 d+b^2 B e-b c (4 B d+9 A e)\right )+\left (16 A c^4 d^3-4 b^4 B e^3+b^3 c e^2 (4 B d+A e)-8 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (5 B d+6 A e)\right ) x\right )}{3 b^4 c^2 \sqrt {b x+c x^2}}+\frac {\left (d (c d-b e) \left (16 A c^3 d^2+4 b^3 B e^2+b^2 c e (B d-A e)-8 b c^2 d (B d+2 A e)\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3 b^4 c^2}-\frac {\left (16 A c^4 d^3-8 b^4 B e^3+b^3 c e^2 (5 B d+2 A e)-8 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (5 B d+4 A e)\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{3 b^4 c^2} \\ & = -\frac {2 (d+e x)^{5/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b c d^2 \left (8 A c^2 d+b^2 B e-b c (4 B d+9 A e)\right )+\left (16 A c^4 d^3-4 b^4 B e^3+b^3 c e^2 (4 B d+A e)-8 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (5 B d+6 A e)\right ) x\right )}{3 b^4 c^2 \sqrt {b x+c x^2}}+\frac {\left (d (c d-b e) \left (16 A c^3 d^2+4 b^3 B e^2+b^2 c e (B d-A e)-8 b c^2 d (B d+2 A e)\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}} \, dx}{3 b^4 c^2 \sqrt {b x+c x^2}}-\frac {\left (\left (16 A c^4 d^3-8 b^4 B e^3+b^3 c e^2 (5 B d+2 A e)-8 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (5 B d+4 A e)\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}} \, dx}{3 b^4 c^2 \sqrt {b x+c x^2}} \\ & = -\frac {2 (d+e x)^{5/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b c d^2 \left (8 A c^2 d+b^2 B e-b c (4 B d+9 A e)\right )+\left (16 A c^4 d^3-4 b^4 B e^3+b^3 c e^2 (4 B d+A e)-8 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (5 B d+6 A e)\right ) x\right )}{3 b^4 c^2 \sqrt {b x+c x^2}}-\frac {\left (\left (16 A c^4 d^3-8 b^4 B e^3+b^3 c e^2 (5 B d+2 A e)-8 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (5 B d+4 A e)\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x}\right ) \int \frac {\sqrt {1+\frac {e x}{d}}}{\sqrt {x} \sqrt {1+\frac {c x}{b}}} \, dx}{3 b^4 c^2 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {\left (d (c d-b e) \left (16 A c^3 d^2+4 b^3 B e^2+b^2 c e (B d-A e)-8 b c^2 d (B d+2 A e)\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}}} \, dx}{3 b^4 c^2 \sqrt {d+e x} \sqrt {b x+c x^2}} \\ & = -\frac {2 (d+e x)^{5/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}+\frac {2 \sqrt {d+e x} \left (b c d^2 \left (8 A c^2 d+b^2 B e-b c (4 B d+9 A e)\right )+\left (16 A c^4 d^3-4 b^4 B e^3+b^3 c e^2 (4 B d+A e)-8 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (5 B d+6 A e)\right ) x\right )}{3 b^4 c^2 \sqrt {b x+c x^2}}-\frac {2 \left (16 A c^4 d^3-8 b^4 B e^3+b^3 c e^2 (5 B d+2 A e)-8 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (5 B d+4 A e)\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} c^{5/2} \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 d (c d-b e) \left (16 A c^3 d^2+4 b^3 B e^2+b^2 c e (B d-A e)-8 b c^2 d (B d+2 A e)\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} c^{5/2} \sqrt {d+e x} \sqrt {b x+c x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 25.17 (sec) , antiderivative size = 530, normalized size of antiderivative = 1.01 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \left (b (d+e x) \left (b (b B-A c) (c d-b e)^3 x^2+(c d-b e)^2 \left (-8 A c^2 d+5 b^2 B e+b c (5 B d-2 A e)\right ) x^2 (b+c x)+A b c^2 d^3 (b+c x)^2+c^2 d^2 (3 b B d-8 A c d+10 A b e) x (b+c x)^2\right )+\sqrt {\frac {b}{c}} x (b+c x) \left (\sqrt {\frac {b}{c}} \left (16 A c^4 d^3-8 b^4 B e^3+b^3 c e^2 (5 B d+2 A e)-8 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (5 B d+4 A e)\right ) (b+c x) (d+e x)+i b e \left (16 A c^4 d^3-8 b^4 B e^3+b^3 c e^2 (5 B d+2 A e)-8 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (5 B d+4 A e)\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-i b e (c d-b e) \left (8 A c^3 d^2+8 b^3 B e^2-b^2 c e (B d+2 A e)-b c^2 d (4 B d+5 A e)\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )\right )}{3 b^5 c^2 (x (b+c x))^{3/2} \sqrt {d+e x}} \]

[In]

Integrate[((A + B*x)*(d + e*x)^(7/2))/(b*x + c*x^2)^(5/2),x]

[Out]

(-2*(b*(d + e*x)*(b*(b*B - A*c)*(c*d - b*e)^3*x^2 + (c*d - b*e)^2*(-8*A*c^2*d + 5*b^2*B*e + b*c*(5*B*d - 2*A*e
))*x^2*(b + c*x) + A*b*c^2*d^3*(b + c*x)^2 + c^2*d^2*(3*b*B*d - 8*A*c*d + 10*A*b*e)*x*(b + c*x)^2) + Sqrt[b/c]
*x*(b + c*x)*(Sqrt[b/c]*(16*A*c^4*d^3 - 8*b^4*B*e^3 + b^3*c*e^2*(5*B*d + 2*A*e) - 8*b*c^3*d^2*(B*d + 3*A*e) +
b^2*c^2*d*e*(5*B*d + 4*A*e))*(b + c*x)*(d + e*x) + I*b*e*(16*A*c^4*d^3 - 8*b^4*B*e^3 + b^3*c*e^2*(5*B*d + 2*A*
e) - 8*b*c^3*d^2*(B*d + 3*A*e) + b^2*c^2*d*e*(5*B*d + 4*A*e))*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*Elli
pticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*b*e*(c*d - b*e)*(8*A*c^3*d^2 + 8*b^3*B*e^2 - b^2*c*e*(B*d
+ 2*A*e) - b*c^2*d*(4*B*d + 5*A*e))*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]
/Sqrt[x]], (c*d)/(b*e)])))/(3*b^5*c^2*(x*(b + c*x))^(3/2)*Sqrt[d + e*x])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1132\) vs. \(2(470)=940\).

Time = 2.28 (sec) , antiderivative size = 1133, normalized size of antiderivative = 2.16

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1133\)
default	\(\text {Expression too large to display}\)	\(3247\)

[In]

int((B*x+A)*(e*x+d)^(7/2)/(c*x^2+b*x)^(5/2),x,method=_RETURNVERBOSE)

[Out]

((e*x+d)*x*(c*x+b))^(1/2)/(x*(c*x+b))^(1/2)/(e*x+d)^(1/2)*(-2/3/b^3*d^3*A*(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2
)/x^2-2/3*(c*e*x^2+b*e*x+c*d*x+b*d)/b^4*d^2*(10*A*b*e-8*A*c*d+3*B*b*d)/(x*(c*e*x^2+b*e*x+c*d*x+b*d))^(1/2)-2/3
*(A*b^3*c*e^3-3*A*b^2*c^2*d*e^2+3*A*b*c^3*d^2*e-A*c^4*d^3-B*b^4*e^3+3*B*b^3*c*d*e^2-3*B*b^2*c^2*d^2*e+B*b*c^3*
d^3)/b^3/c^4*(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)/(x+b/c)^2+2/3*(c*e*x^2+c*d*x)*(2*A*b^3*c*e^3+4*A*b^2*c^2*d*
e^2-14*A*b*c^3*d^2*e+8*A*c^4*d^3-5*B*b^4*e^3+5*B*b^3*c*d*e^2+5*B*b^2*c^2*d^2*e-5*B*b*c^3*d^3)/c^3/b^4/((x+b/c)
*(c*e*x^2+c*d*x))^(1/2)+2*(e^3*(A*c*e-2*B*b*e+4*B*c*d)/c^3-1/3*d^3/b^3*A*c*e-1/3*(A*b^3*c*e^3-3*A*b^2*c^2*d*e^
2+3*A*b*c^3*d^2*e-A*c^4*d^3-B*b^4*e^3+3*B*b^3*c*d*e^2-3*B*b^2*c^2*d^2*e+B*b*c^3*d^3)/c^3*e/b^3-1/3*(2*A*b^3*c*
e^3+4*A*b^2*c^2*d*e^2-14*A*b*c^3*d^2*e+8*A*c^4*d^3-5*B*b^4*e^3+5*B*b^3*c*d*e^2+5*B*b^2*c^2*d^2*e-5*B*b*c^3*d^3
)/c^3*(b*e-c*d)/b^4-1/3/c^2*d*(2*A*b^3*c*e^3+4*A*b^2*c^2*d*e^2-14*A*b*c^3*d^2*e+8*A*c^4*d^3-5*B*b^4*e^3+5*B*b^
3*c*d*e^2+5*B*b^2*c^2*d^2*e-5*B*b*c^3*d^3)/b^4)*b/c*((x+b/c)/b*c)^(1/2)*((x+d/e)/(-b/c+d/e))^(1/2)*(-c*x/b)^(1
/2)/(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)*EllipticF(((x+b/c)/b*c)^(1/2),(-b/c/(-b/c+d/e))^(1/2))+2*(B*e^4/c^2+
1/3*c*e*d^2*(10*A*b*e-8*A*c*d+3*B*b*d)/b^4-1/3*(2*A*b^3*c*e^3+4*A*b^2*c^2*d*e^2-14*A*b*c^3*d^2*e+8*A*c^4*d^3-5
*B*b^4*e^3+5*B*b^3*c*d*e^2+5*B*b^2*c^2*d^2*e-5*B*b*c^3*d^3)/c^2*e/b^4)*b/c*((x+b/c)/b*c)^(1/2)*((x+d/e)/(-b/c+
d/e))^(1/2)*(-c*x/b)^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)*((-b/c+d/e)*EllipticE(((x+b/c)/b*c)^(1/2),(-b
/c/(-b/c+d/e))^(1/2))-d/e*EllipticF(((x+b/c)/b*c)^(1/2),(-b/c/(-b/c+d/e))^(1/2))))

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.18 (sec) , antiderivative size = 1266, normalized size of antiderivative = 2.42 \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\text {Too large to display} \]

[In]

integrate((B*x+A)*(e*x+d)^(7/2)/(c*x^2+b*x)^(5/2),x, algorithm="fricas")

[Out]

-2/9*(((8*(B*b*c^6 - 2*A*c^7)*d^4 - (9*B*b^2*c^5 - 32*A*b*c^6)*d^3*e - (4*B*b^3*c^4 + 13*A*b^2*c^5)*d^2*e^2 -
3*(3*B*b^4*c^3 + A*b^3*c^4)*d*e^3 + 2*(4*B*b^5*c^2 - A*b^4*c^3)*e^4)*x^4 + 2*(8*(B*b^2*c^5 - 2*A*b*c^6)*d^4 -
(9*B*b^3*c^4 - 32*A*b^2*c^5)*d^3*e - (4*B*b^4*c^3 + 13*A*b^3*c^4)*d^2*e^2 - 3*(3*B*b^5*c^2 + A*b^4*c^3)*d*e^3
+ 2*(4*B*b^6*c - A*b^5*c^2)*e^4)*x^3 + (8*(B*b^3*c^4 - 2*A*b^2*c^5)*d^4 - (9*B*b^4*c^3 - 32*A*b^3*c^4)*d^3*e -
(4*B*b^5*c^2 + 13*A*b^4*c^3)*d^2*e^2 - 3*(3*B*b^6*c + A*b^5*c^2)*d*e^3 + 2*(4*B*b^7 - A*b^6*c)*e^4)*x^2)*sqrt
(c*e)*weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*b^
2*c*d*e^2 + 2*b^3*e^3)/(c^3*e^3), 1/3*(3*c*e*x + c*d + b*e)/(c*e)) + 3*((8*(B*b*c^6 - 2*A*c^7)*d^3*e - (5*B*b^
2*c^5 - 24*A*b*c^6)*d^2*e^2 - (5*B*b^3*c^4 + 4*A*b^2*c^5)*d*e^3 + 2*(4*B*b^4*c^3 - A*b^3*c^4)*e^4)*x^4 + 2*(8*
(B*b^2*c^5 - 2*A*b*c^6)*d^3*e - (5*B*b^3*c^4 - 24*A*b^2*c^5)*d^2*e^2 - (5*B*b^4*c^3 + 4*A*b^3*c^4)*d*e^3 + 2*(
4*B*b^5*c^2 - A*b^4*c^3)*e^4)*x^3 + (8*(B*b^3*c^4 - 2*A*b^2*c^5)*d^3*e - (5*B*b^4*c^3 - 24*A*b^3*c^4)*d^2*e^2
- (5*B*b^5*c^2 + 4*A*b^4*c^3)*d*e^3 + 2*(4*B*b^6*c - A*b^5*c^2)*e^4)*x^2)*sqrt(c*e)*weierstrassZeta(4/3*(c^2*d
^2 - b*c*d*e + b^2*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 2*b^3*e^3)/(c^3*e^3), we
ierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*b^2*c*d*e^
2 + 2*b^3*e^3)/(c^3*e^3), 1/3*(3*c*e*x + c*d + b*e)/(c*e))) + 3*(A*b^3*c^4*d^3*e + (8*(B*b*c^6 - 2*A*c^7)*d^3*
e - (5*B*b^2*c^5 - 24*A*b*c^6)*d^2*e^2 - (5*B*b^3*c^4 + 4*A*b^2*c^5)*d*e^3 + (5*B*b^4*c^3 - 2*A*b^3*c^4)*e^4)*
x^3 + (12*(B*b^2*c^5 - 2*A*b*c^6)*d^3*e - (8*B*b^3*c^4 - 37*A*b^2*c^5)*d^2*e^2 - (2*B*b^4*c^3 + 7*A*b^3*c^4)*d
*e^3 + (4*B*b^5*c^2 - A*b^4*c^3)*e^4)*x^2 + (10*A*b^3*c^4*d^2*e^2 + 3*(B*b^3*c^4 - 2*A*b^2*c^5)*d^3*e)*x)*sqrt
(c*x^2 + b*x)*sqrt(e*x + d))/(b^4*c^6*e*x^4 + 2*b^5*c^5*e*x^3 + b^6*c^4*e*x^2)

Sympy [F(-1)]

Timed out. \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate((B*x+A)*(e*x+d)**(7/2)/(c*x**2+b*x)**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {7}{2}}}{{\left (c x^{2} + b x\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((B*x+A)*(e*x+d)^(7/2)/(c*x^2+b*x)^(5/2),x, algorithm="maxima")

[Out]

integrate((B*x + A)*(e*x + d)^(7/2)/(c*x^2 + b*x)^(5/2), x)

Giac [F]

\[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {7}{2}}}{{\left (c x^{2} + b x\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((B*x+A)*(e*x+d)^(7/2)/(c*x^2+b*x)^(5/2),x, algorithm="giac")

[Out]

integrate((B*x + A)*(e*x + d)^(7/2)/(c*x^2 + b*x)^(5/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(A+B x) (d+e x)^{7/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^{7/2}}{{\left (c\,x^2+b\,x\right )}^{5/2}} \,d x \]

[In]

int(((A + B*x)*(d + e*x)^(7/2))/(b*x + c*x^2)^(5/2),x)

[Out]

int(((A + B*x)*(d + e*x)^(7/2))/(b*x + c*x^2)^(5/2), x)